Electric Current, Drift Velocity and Current Density

1Instant light, slow electrons

Flick the switch by the classroom door and the lamp on the ceiling lights at once. You might think electrons rush from the switch to the lamp at enormous speed. In fact, as you will calculate in this lesson, the electrons in the copper wire move along at less than a millimetre per second, slower than a snail. Resolving that puzzle means looking inside the wire at the charge carriers and how they move: their drift velocity.

2Electric current

An electric current is a flow of charge. The current $II$ through a cross-section of a conductor is the rate at which charge passes it:

The SI unit is the ampere (A); 1 A is a flow of one coulomb per second. In metals the moving charges are free electrons, each carrying charge $-e=-1.60\times {10}^{-19}\mathrm{C}-e\; =\; -1.60\; \backslash times\; 10^\{-19\}\backslash ,\backslash mathrm\{C\}$; in electrolytes and gases they are ions; in semiconductors, electrons and "holes". By convention the direction of current is the direction in which positive charge would move, so in a metal wire the electrons drift the opposite way to the conventional current.

3The drift velocity equation

The free electrons in a metal move randomly at high speed (about ${10}^5\mathrm{m}{\mathrm{s}}^{-1}10^\{5\}\backslash ,\backslash mathrm\{m\backslash ,s^\{-1\}\}$) even with no current, but their random motions cancel out. When a potential difference is applied, an electric field in the wire pushes them, and on top of the random motion they acquire a small average velocity along the wire, the drift velocity $vv$.

Figure 1.1: All the carriers in the shaded slice, of length $v\mathrm{\Delta }tv\backslash Delta\; t$, cross the end face in time $\mathrm{\Delta }t\backslash Delta\; t$.

Let the wire have cross-sectional area $AA$ and contain $nn$ free charge carriers per cubic metre, each of charge $qq$. In time $\mathrm{\Delta }t\backslash Delta\; t$ each carrier moves a distance $v\mathrm{\Delta }tv\backslash Delta\; t$, so all the carriers in a slice of volume $Av\mathrm{\Delta }tAv\backslash Delta\; t$ pass through a cross-section. They carry charge $\mathrm{\Delta }Q=nAv\mathrm{\Delta }tq\backslash Delta\; Q\; =\; nAv\backslash Delta\; t\backslash ,q$. Dividing by $\mathrm{\Delta }t\backslash Delta\; t$:

For a given current, the drift speed is large where $nn$ or $AA$ is small. Copper has about one free electron per atom, $n\approx 8.5\times {10}^{28}{\mathrm{m}}^{-3}n\; \backslash approx\; 8.5\; \backslash times\; 10^\{28\}\backslash ,\backslash mathrm\{m^\{-3\}\}$, so its drift speeds are tiny. A semiconductor such as silicon has far fewer carriers, so the same current needs much faster drift.

4So why does the lamp light at once?

The wire is already full of free electrons, all along its length. When the switch closes, the electric field is set up throughout the circuit almost at the speed of light, and every free electron in the circuit, including those already in the lamp's filament, starts drifting at practically the same moment. It is like a garden hose that is already full of water: when you open the tap, water comes out of the far end immediately, although no drop of water has travelled the length of the hose. The energy is carried by the field; the electrons only need to move a tiny distance.

Same current, different wires

In a series circuit the current is the same at every point (charge cannot pile up). If the wire narrows, $nAvnAv$ must stay constant, so the electrons drift faster in the thin part and the current density there is larger (Figure 1.2). A high current density heats a wire strongly, which is why thin wires cannot safely carry large currents and why fuse wires are thin.

Figure 1.2: One current, two wires. The thin wire has four times the drift speed and current density.

5Experiment 1.1: watching charge carriers drift

Apparatus: a microscope slide, a strip of filter paper, salt solution, two crocodile clips, a 12 V d.c. supply, a tiny crystal of potassium permanganate, tweezers, gloves.

Figure 1.3: Ions from the crystal carry part of the current through the damp paper.

1. Soak the filter paper in salt solution and lay it on the slide. Clip each end and connect the clips to the supply.
2. Using tweezers, place one small crystal in the middle of the paper. Switch on.
3. Watch the purple colour over 10–15 minutes and measure how far its front moves.

The purple permanganate ions are negative, so they drift towards the positive clip, a few millimetres in several minutes: the carriers move slowly even though the current starts at once. In a metal the carriers are invisible electrons, but they behave in the same way.